Abstract
A lattice gas with non-conserved spin flip dynamics (of both non-Glauber and Glauber types) is considered at T≪T c , the critical temperature. For arbitrary supersaturation, S, a general expression for the inverse of the nucleation rate along the lowest energy path is derived. The exponential part is identical to the one by Neves and Schonmann [Commun. Math. Phys. 137:20 (1991)]. The preexponential can be expressed in terms of elliptic theta-functions for small S, and in the limits, respectively, of S≫T/φ or S≪T/φ (−φ being the nearest-neighbor interaction energy), elementary versions of the general expression are further obtained. The preexponential has a smooth component, as well as small-scale modulations which are approximately periodic in the inverse supersaturation. For S≪T/φ, the smooth part is proportional to \(\sqrt S \), in contrast to the zero-T limit where it is linear in S. The latter limit becomes apparent only at extremely low temperatures which are cubic in S.
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REFERENCES
F. F. Abraham, Homogeneous Nucleation Theory (Academic Press, New York, 1974).
A. A. Chernov, Modern Crystallography III, Crystal Growth (Springer, Berlin, 1984).
P. Debenedetti, Metastable Liquids (Princeton, New Jersey, 1996).
A. D. Linde, Uspekhi Fisicheskih Nauk 144:177(1984).
M. Volmer and A. Weber, Z. Phys. Chem. 119:227(1926); L. Farkas, ibid., 125:236(1927); R. Becker and W. Döring, Ann. Phys. 24:719(1935); J. Frenkel, Kinetic Theory of Liquids (Oxford University, Oxford, 1946).
J. Lothe and G. Pound, J. Chem. Phys. 36:2080(1962).
H. Reiss, J. L. Katz, and E. Cohen, J. Chem. Phys. 48:5553(1968).
C. Rottman and M. Wortis, Phys. Rev. B 24:6274(1981); R. K. P. Zia and J. E. Avron, Phys. Rev. B 25:2042(1982).
R. K. P. Zia, J. Stat. Phys. 45:801(1986); V. A. Shneidman and R. K. P. Zia, Phys. Rev. B 63:085410(2001).
M. E. Fisher, Phys. 3:255(1967).
J. S. Langer, Ann. Phys. (N.Y.) 65:53(1971).
N. J. Günther, D. A. Nicole, and D. J. Wallace, J. Phys. A 13:1755(1980).
J. D. Gunton, M. San Miguel, and P. S. Sahni, in Phase Transitions and Critical Phenomena, Vol. 8, C. Domb and J. L. Lebowitz, ed. (Academic, New York, 1983), p. 267.
E. Stoll, K. Binder, and T. Schneider, Phys. Rev. B 6:2777(1972); K. Binder and D. Stauffer, Adv. Physics 25:343(1976).
P. A. Rikvold, H. Tomita, S. Miyashita, and S. W. Sides, Phys. Rev. E 49:5080(1994).
C. C. Günther, P. A. Rikvold, and M. A. Novotny, Phys. Rev. Lett. 71:3898(1993); C. C. Günther, P. A. Rikvold, and M. A. NovotnyPhys. A 212:194(1994).
V. A. Shneidman, K. A. Jackson, and K. M. Beatty, J. Chem. Phys. 111:6932(1999).
E. J. Neves and R. H. Schonmann, Commun. Math. Phys. 137:209(1991).
R. Kotecky and E. Olivieri, J. Stat. Phys. 75:409(1994).
M. A. Novotny, Low-temperature metastable lifetimes of the square-lattice Ising ferromagnet, in Springer Proc. in Physics, Vol. 82: Computer Simulation Studies in Condensed-Matter Physics IX, D. P. Landau, K. K. Mon, and H.-B. Schüttler, eds. (Springer, Berlin, 1997), p. 182.
A. Bovier and F. Manzo, J. Stat. Phys. 107:757(2002).
K. Park and M. A. Novotny, cond-mat/0109214 (2001).
M. A. Novotny, cond-mat/0108429 (2001).
V. A. Shneidman and G. Nita, Phys. Rev. Lett. 89:25701(2002).
M. A. Novotny, cond-mat/0204098 (2002).
R. J. Baxter. Exactly Solved Models in Statistical Mechanics, Chap. 1 (Academic Press, New York, 1982).
M. Kalos, J. L. Lebowitz, O. Penrose, and A. Sur, J. Stat. Phys. 18:39(1978).
O. Penrose and J. L. Lebowitz, in Studies in Statistical Mechanics, Vol. VII, Fluctuation Phenomena, E. Montrol and J. L. Lebowitz, eds. (North-Holland, Amsterdam, 1979); J. Ball, J. Carr, and O. Penrose, Commun. Math. Phys. 104:657(1986).
V. A. Shneidman, K. A. Jackson, and K. M. Beatty, J. Cryst. Growth 212:564(2000).
J. P. Marchand and P. A. Martin, Phys. A 127:681(1984).
Ph. A. Martin, J. Stat. Phys. 16:149(1977).
O. Penrose, Commun. Math. Phys. 124:515(1989).
V. A. Shneidman, Sov. Phys. Tech. Phys. 32:76(1987); ibid. 33:1338(1988).
V. A. Shneidman, J. Chem. Phys. 115:8141(2001).
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
V. A. Shneidman and G. M. Nita, in preparation.
S. B. Rutkevich, J. Stat. Phys. 104:589(2001).
P. A. Rikvold and M. A. Novotny, private communication.
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Shneidman, V.A. On the Lowest Energy Nucleation Path in a Supersaturated Lattice Gas. Journal of Statistical Physics 112, 293–318 (2003). https://doi.org/10.1023/A:1023687822656
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DOI: https://doi.org/10.1023/A:1023687822656